Zivot - Lectures on Structural Change.pdf

7/27/2019 Zivot - Lectures on Structural Change.pdf

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Lectures on Structural Change

Eric Zivot

Department of Economics, University of Washington

April 5, 2003

1 Overview of Testing for and Estimating Struc-

tural Change in Econometric Models

1. Day 1: Tests of Parameter Constancy

2. Day 2: Estimation of Models with Structural Change

3. Day 3: Time Varying Parameter Models

2 Some Preliminary Asymptotic Theory

Reference: Stock, J.H. (1994) Unit Roots, Structural Breaks and Trends,in Handbook of Econometrics, Vol. IV.

3 Tests of Parameter Constancy in Linear Mod-

els

3.1 Motivation

Diagnostics for model adequacy

Provide information about out-of-sample forecasting accuracy

Within-sample parameter constancy is a necessary condition for super-exogeneity

3.2 Example Data Sets

3.2.1 Simulated Data

Consider the linear regression model

yt = + xt + t, t = 1, . . . , T = 200

xt iid N(0, 1)t iid N(0, 2)

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No structural change parameterization: = 0, = 1, = 0.5Structural change cases

Break in intercept: = 1 for t > 100

Break in slope: = 3 for t > 100

Break in error variance: = 0.25 for t > 100

Random walk in slope: = t = t1 + t, t iid N(0, 0.1) and 0 = 1.

(show simulated data)

3.2.2 US/DM Monthly Exchange rate data

Let

st = log of spot exchange rate in month t

ft = log of forward exchange rate in month t

The forward rate unbiased hypothesis is typically investigated using the so-calleddifferences regression

st+1 = + (ft st) + t+1ft st = iUSt iDMt = forward discount

If the forward rate ft is an unbiased forecast of the future spot rate st+1 thenwe should find

= 0 and = 1

The forward discount is often modeled as an AR(1) model

ft st = + (ft1 st1) + utStatistical Issues

st+1 is close to random walk with large variance

ft st behaves like highly persistent AR(1) with small variance ft st appears to be unstable over time

3.3 Chow Forecast Test

Reference: Chow, G.C. (1960). Tests of Equality between Sets of Coeffi-cients in Two Linear Regressions, Econometrica, 52, 211-22.

Consider the linear regression model with k variables

yt = x0

t + ut, ut (0, 2), t = 1, . . . , ny = X + u

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Parameter constancy hypothesis

H0 : is constant

Intuition

If parameters are constant then out-of-sample forecasts should be unbiased(forecast errors have mean zero)

Test construction:

Split sample into n1 > k and n2 = n n1 observations

y =

y1y2

n1n2

, X =

X1X2

n1n2

Fit model using first n1

observations

1 = (X0

1X1)1X01y1

u1 = y1 X1121 = u

0

1u1/(n1 k)

Use 1 and X2 to predict y2 using next n2 observations

y2 = X21

Compute out-of-sample prediction errors

u2 = y2y2 = y2X21

Under H0 : is constant

u2 = u2 X2(1)

and

E[u2] = 0

var(u2) = 2

In2 + X2(X0

1X1)1X02

Further, If the errors u are Gaussian then

u2 N(0,var(u2))u02var(u2)

1u2 2(n2)(n1 k)21/2 2(n1 k)

This motivates the Chow forecast test statistic

ChowFCST(n2) =u02

In2 + X2(X

0

1X1)1X02

u2

n221

F(n2, n1 k)

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Decision: Reject H0 at 5% level if

ChowFCST(n2) > cv0.05

Remarks:

Test is a general specification test for unbiased forecasts

Popular with LSE methodology

Implementation requires a priori splitting of data into fit and forecastsamples

3.3.1 Application: Simulated Data

Chow Forecast Test

n2Model 100 50 25No SC 1.121 1.189 1.331Mean shift 9.130*** 1.329* 1.061Slope shift 9.055*** 2.067*** 1.545*Var shift 0.568 0.726 0.864RW slope 2.183*** 1.302 0.550

3.4 CUSUM and CUSUMSQ Tests

Reference: Brown, R.L., J. Durbin and J.M. Evans (1975). Techniquesfor Testing the Constancy of Regression Relationships over Time, Journal ofthe Royal Statistical Society, Series B, 35, 149-192.

3.4.1 Recursive least squares estimation

The recursive least squares (RLS) estimates of are based on estimating

yt = 0

txt + t, t = 1, . . . , n

by least squares recursively for t = k + 1, . . . , n giving nk least squares (RLS)estimates (k+1, . . . , T).

RLS estimates may be efficiently computed using the Kalman Filter

If is constant over time then t should quickly settle down near a com-mon value.

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3.4.2 Recursive residuals

Formal tests for structural stability of the regression coefficients may be com-puted from the standardized 1 step ahead recursive residuals

wt =vt

ft=

yt 0

t1xtft

ft = 2h

1 + x0t(X0

tXt)1xt

iIntuition:

Ifi changes in the next period then the forecast error will not have meanzero

wt are recursive Chow Forecast t-statistics with n2 = 1

3.4.3 CUSUM statistic

The CUSUM statistic of Brown, Durbin and Evans (1975) is

CUSUMt =tX

j=k+1

wjw

2w =1

n knXt=1

(wt w)2

Under the null hypothesis that is constant, CUSUMt has mean zero andvariance that is proportional to t

k

1.

3.4.4 CUSUMSQ statistic

THE CUSUMSQ statistic is

CUSUMSQt =

Ptj=k+1 w

2jPn

j=k+1 w2j

Under the null that is constant, CUSUMSQt behaves like a 2(t) and con-

fidence bounds can be easily derived.


(insert graphs here)

Remarks

Ploberger and Kramer (1990) show the CUSUM test can be constructedwith OLS residuals instead of recursive residuals

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CUSUM Test is essentially a test to detect instability in intercept alone

CUSUM Test has power only in direction of the mean regressors

CUSUMSQ has power for changing variance

There are tests with better power

3.4.6 Application: Exchange Rate Regression


3.5 Nybloms Parameter Stability Test

Reference:Nyblom, J. (1989). Testing for the Constancy of Parameters OverTime, Journal of the American Statistical Association, 84 (405), 223-230.


yt = x0

t + t, t = 1, . . . , n

The time varying parameter (TVP) alternative model assumes

= t = t1 + t, it (0, 2i

), i = 1, . . . , k

The hypotheses of interest are

H0 : is constant 2i

= 0 for all i

H1 : 2i

> 0 for some i

Nyblom (1989) derives the locally best invariant test as the Lagrange multiplier

test. The score assuming Gaussian errors isnXt=1

xtt = 0

t = yt x0t= (X0X)1X0y

Define

ft = xtt

St =t

Xj=1ft = cumulative sums

V = n1X0X

Note thatnXj=1

ft = 0

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Nyblom derives the LM statistic

L = 1n2

nXt=1

StV1St

=1

n2tr

"V1

nXt=1

StS0

t

#

Under mild assumptions regarding the behavior of the regressors, the limitingdistribution of L under the null is a Camer-von Mises distribution:

L Z10

Bk()Bk()

0d

Bk

() = Wk() Wk(1)W

k() = k dimensional Brownian motion


L > cv0.05

Remarks:

Distribution of L is non-standard and depends on k.

Critical values are computed by simulation and are given in Nyblom,Hansen (1992) and Hansen (1997)

Test is for constancy of all parameters

Test is not informative about the date or type of structural change

Test is applicable for models estimated by methods other than OLS

Distribution of L is different if xt is non-stationary (unit root, determin-istic trend). See Hansen (1992).


Nyblom TestModel LcNo SC .332Mean shift 13.14

Slope shift 14.13

var shift .351RW slope 9.77

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3.5.2 Application: Exchange rate regression

Nyblom TestModel LcAR(1) 1.27

Diff reg .413

3.6 Hansens Parameter Stability Tests

References

1. Hansen, B.E. (1992). Testing for Parameter Instability in Linear Mod-els Journal of Policy Modeling, 14(4), 517-533.

2. Hansen, B.E. (1992). Tests for Parameter Instability in Regressionswith I(1) Processes, Journal of Business and Economic Statistics, 10,321-336.

Idea: Extension of Nybloms LM test to individual coefficients.Under the null of constant parameters, the score vector from the linear model

with Gaussian errors is

nXt=1

xitt = 0, i = 1, . . . , kX(2t 2) = 0

t = yt x0t

2 = n1n

Xt=1 2tDefine

fit =

xitt

2t 2i = 1, . . . ki = k + 1

Sit =tX

j=1

fij , i = 1, . . . , k + 1

Note thatn

Xi=1fit = 0, i = 1, . . . , k + 1

3.6.1 Individual Coefficient Tests

Hansens LM test for

H0 : i is constant, i = 1, . . . , k

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and forH0 :

2 is constant

is

Li =1

nVi

nXt=1

S2it, i = 1, . . . , k

Vi =nXt=1

f2it

Under H0 : i is constant or H0 : 2 is constant

Li Z10

B1 ()B1 ()d


Li > cv0.05 = 0.470

3.6.2 Joint Test for All Coefficients

For testing the joint hypothesis

H0 : and 2 are constant

define the (k + 1) 1 vectors

ft = (f1t, . . . , f k+1,t)0

St = (S1t, . . . , S k+1,t)0

Hansens LM statistic for testing the constancy of all parameters is

Lc =1

n

nXt=1

S0tV1St =

1

ntr

V1

nXt=1

StS0

t

!

V =nXt=1

ftf0

t

Under the null of no-structural change

Lc

Z1

0

Bk+1()Bk+1()d


Lc > cv0.05

Remarks

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Tests are very easy to compute and are robust to heteroskedasticity

Null distribution is non-standard and depends upon number of parameterstested for stability

Individual tests are informative about the type of structural change

Tests are not informative about the date of structural change

Hansens L1 test for constancy of intercept is analogous to the CUSUMtest

Hansens Lk+1 test for constancy of variance is analogous to CUSUMSQtest

Hansens Lc test for constancy of all parameters is similar to Nybolomstest

Distribution of tests is different if data are nonstationary (unit root, de-terministic trend) - see Hansen (1992), JBES.


Hansen TestsModel 2 JointNo SC .179 .134 .248 .503Mean shift 13.19 .234 .064 13.3

Slope shift .588 5.11 .067 5.25

var shift .226 .119 .376 .736RW slope .253 4.08 .196 4.4

3.6.4 Application: Exchange rate regression contd

Hansen TestsModel intercept slope variance JointAR(1) .382 .147 2.94 3.90

Diff reg .104 .153 .186 .520

4 Tests for Single Structural Change


yt = x0

tt + t, t = 1 . . . , n

No structural change null hypothesis

H0 : t=

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Single break date alternative hypothesis

H1 : t= , t m = break datet = + , t > m and 6= 0k < m < n k =

m

n= break fraction

Remarks:

Under no break null = 0.

Pure structural change model: all coefficients change (i 6= 0 for i =1, . . . , k)

Partial structural change model: some coefficients change (i 6= 0 for somei)

m = [ n], [] = integer part

4.1 Chows Test with Known Break Date

Assume: m or is knownFor a data interval [r , . . . , s] such that s r > k define

r,s = OLS estimate of

r,s = OLS residual vector

SS Rr,s = 0

r,sr,s = sum of squared residuals

Chows breakpoint test for testing H0 vs. H1 with m known is

Fn

mn

= Fn () =

(SS R1,n (SS R1,m + SS Rm+1,n))/k(SS R1,m + SS Rm+1,n) /(n 2k)

The Chow test may also be computed as the F-statistic for testing = 0 fromthe dummy variable regression

yt = x0

t + Dt(m)x0

t+ t

Dt(m) = 1 if t > m; 0 otherwise

Under H0 : = 0 with m known

Fn() F(k, n 2k)

k Fn()d

2(k)Decision: Reject H0 at 5% level if

Fn() > F0.95(k, n k)k Fn() >

20.95(k)

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Chow Breakpoint TestF200(0.5) F200(0.25) F200(0.75)No SC 0.808 0.081 1.55Mean shift 377*** 13.03*** 11.21***Slope shift 374*** 10.97*** 17.57***Var shift 1.071 0.117 1.204RW slope 80.14*** 4.058** 2.218

4.2 Quandts LR Test with Unknown Break Date

References:

1. Quandt, R.E. (1960). Tests of Hypotheses that a Linear System Obeys

Two Separate Regimes, Journal of the American Statistical Association,55, 324-330.

2. Davies, R.A. (1977). Hypothesis Testing When a Nuisance Parameteris Present only Under the Alternative, Biometrika, 64, 247-254.

3. Kim, H.-J., and D. Siegmund (1989). The Likelihood Ratio Test fora Change-Point in Simple Linear Regression, Biometrika, 76, 3, 409-23.

4. Andrews, D.W.K. (1993). Tests for Parameter Instability and Struc-tural Change with Unknown Change Point, Econometrica, 59, 817-858.

5. Hansen, B.E. (1997). Approximate Asymptotic P Values for Structural-Change Tests, Journal of Business and Economic Statistics, 15, 60-67.

Assume: m or is unknown.Quandt considered the LR statistic for testing H0 : = 0 vs. H1 : 6= 0

when m is unknown. This turns out to be the maximal Fn() statistic over arange of break dates m0, . . . , m1 :

QLR = maxm[m0,m1]

Fn

mn

= max

[0,1]Fn()

i =min

= trimming parameters, i = 0, 1

Remarks

QLR is also know as Andrews supF statistic

Trimming parameters 0 and 1 must be set

Cannot have 0 = 1 and 1 = 1 because breaks are hard to identifynear beginning and end of sample

Information about location of break can be used to specify 0 and1

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Andrews recommends 0 = 0.15 and 1 = 0.85 if there is no knowl-edge of break date

Implicitly, the break data m and break fraction are estimated using

m = arg maxm

Fn

mn

= m/n

Under the null, m defined under the alternative is not identified. This isan example of the Davies problem.

Davies (1977) showed that if estimated parameters are unidentified underthe null, standard 2 inference does not obtain.

Under H0 : = 0, Kim and Siegmund (1989) showed

k QLR sup[0,1]

Bk()0Bk()

(1 )Bk() = Wk() Wk(1) = Brownian Bridge


k QLR > cv0.05

Remarks

Distribution of QLR is non-standard and depends on the number of vari-ables k and the trimming parameters 0 and 1

Critical values for various values of 0 and 1 computed by simulation aregiven in Andrews (1993), and are larger than 2(k) critical values. For0 = 0.15 and 1 = 0.85

5% critical valuesk 2(k) QLR k QLR1 3.84 8.8510 18.3 27.03

P-values can be computed using techniques from Hansen (1997)

Graphical plot of Fn() statistics is informative to locate the break date

4.2.1 Application: Simulated data

QLR or sup-F Test

QLR

bm

b

No SC 2.87 142 0.71Mean shift 377*** 101 0.51Slope shift 374*** 101 0.51Var shift 2.36 142 0.71RW slope 113*** 77 0.39


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4.2.2 Application: Exchange rate data

QLR or sup-F TestQLR bm b

AR(1) 12.13*** 1989:05 0.65Diff reg 4.08 1991:03 0.74

4.3 Optimal Tests with Unknown Break Date

References:

1. Andrews, D.W.K. and W. Ploberger (1994). Optimal Tests Whena Nuisance Parameter Is Present Only Under the Alternative, Economet-rica, 62, 1383-1414.

Andrews and Ploberger (1994) derive tests for structural change with anunknown break date with optimal power. These tests turn out to be weightedaverages of the Chow breakpoint statistics Fn(

mn

) used to compute the QLRstatistic:

ExpFn = ln

1

m2 m1 + 1m2X

t=m1

exp

1

2k Fn

t

n

!

AveFn =1

m2 m1 + 1m2Xt=m1

k Fn

t

n

k = number of regressors being tested

Remarks

Asymptotic null distributions are non-standard and depend on k, 0 and1

Critical values are given in Andrews and Ploberger; P-values can be com-puted using techniques of Hansen (1997)

Tests can have higher power than QLR statistic

Tests are not informative about location of break date

4.4 Empirical Application

Reference: Stock and Watson (199?), Journal of Business and Economic

Statistics.

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Zivot - Lectures on Structural Change.pdf